# An isosceles triangle has sides A, B, and C, such that sides A and B have the same length. Side C has a length of 8  and the triangle has an area of 32 . What are the lengths of sides A and B?

Dec 10, 2016

$\sqrt{80}$

#### Explanation:

Area of a triangle $A = \frac{1}{2} \cdot b \cdot h$, where $b$ is the base and $h$ is the height.

Given that $A = B$, base $C = 8$, and area of the triangle $= 32$,
$\implies 32 = \frac{1}{2} \cdot 8 \cdot h$
$\implies h = 8$

${A}^{2} = {h}^{2} + {\left(\frac{C}{2}\right)}^{2}$

$\implies A = \sqrt{{8}^{2} + {\left(\frac{8}{4}\right)}^{2}} = \sqrt{80}$

Hence, $A = B = \sqrt{80}$